Following a standard approach (see, e.g., Chapter 3 in [1]}), we introduce a
probability measure \(\tilde{\mathbb {P}}\) 
by specifying its Radon–Nikodym derivative with respect to \(\mathbb {P}\) 
to be given by \(Z_t(X,Y)^{-1}\) , i.e.
\(\left. \frac{{\mathrm {d}} \tilde{\mathbb {P}}}{{\mathrm {d}} \mathbb {P}}\right| _{\mathcal {G}_t}=Z_t.\) 
